# Rings

Learn the rings and their residue classes. Furthermore, we’ll also learn about units and zero-divisors in this lesson.

## Introduction

We’ve introduced groups as an abstract structure that contains a set of elements together with a binary operation, whereas some specific group axioms have to be satisfied. On the contrary to a group $G$, a **ring** $R$ contains two mathematical operations: an addition $+$ and a multiplication $\cdot$. A ring is defined as an abstract structure that maintains an abelian group structure under the addition operation but not necessarily under multiplication.

## Definition of a ring

A set of elements $R$ together with both operations

$\begin{aligned} &+ \space : R \times R \rightarrow R,(a, b) \mapsto a+b, \quad \text { and } \\ &\cdot \space: R \times R \rightarrow R,(a, b) \mapsto a \cdot b \end{aligned}$

is called a ring, if the following properties are fulfilled:

- R1: $R$ is an abelian group under addition $(+)$.
- R2: The multiplication $\cdot$ is associative.
- R3: The multiplication $(\cdot)$ is distributive with respect to the addition $(+)$, i.e., for all $a, b, c \in R$ :
$a \cdot(b+c)=a \cdot b+a \cdot c \text { and }(a+b) \cdot c=a \cdot c+b \cdot c.$

The neutral element $0 \in R$ of the addition is called the **zero elements of** $R$.

Note:We usually write $(R,+, \cdot)$ to $R$. The requirements for multiplication are weaker than the ones for addition because the multiplication operation doesn’t have to be necessarily commutative. There’s also no requirement for the existence of a multiplicative neutral element $1$.

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